\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 19:47:57} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - D\U{f4}kazy\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Aplik\'{a}cie deriv\'{a}ci\'{\i}} \subsection{D\^{o}kaz Taylorovej vety} \textbf{D\^{o}kaz: }Nech \[ g:\left\langle a,b\right\rangle \longrightarrow \mathbf{R},g\left( x\right) =f(b)-f(x)-f\,\/\,^{\prime }(x)(b-x)-\frac{f\,\/\,^{\prime \prime }\left( x\right) }{2!}\left( b-x\right) ^{2}-...-\frac{f^{\left( n\right) }\left( x\right) }{n!}\left( b-x\right) ^{n}-\lambda \frac{\left( b-x\right) ^{n+1}}{% \left( n+1\right) !}. \]% Plat\'{\i}: $g(b)=0$. Zvol\'{\i}me si $\lambda $ tak, aby aj $g(a)=0$ (d\'{a} sa to urobi\v{t}). Potom $g$ je spojit\'{a} funkcia na $\left\langle a,b\right\rangle $ a diferencovate\v{l}n\'{a} na $(a,b)$. Z Rolleovej vety plynie, \v{z}e $\exists c\in \left( a,b\right) $ tak\'{e}, \v{z}e $g^{\prime }(c)=0$. Plat\'{\i}: \[ g^{\prime }\left( x\right) =-f\,\/\,^{\prime }(x)-f\,\/\,^{\prime \prime }(x)(b-x)-\frac{f\,\/\,^{\prime \prime \prime }\left( x\right) }{2!}\left( b-x\right) ^{2}-...-\frac{f^{\left( n+1\right) }\left( x\right) }{n!}\left( b-x\right) ^{n}+ \]% \[ +f\,\/\,^{\prime }(x)+f\,\/\,^{\prime \prime }\left( x\right) \left( b-x\right) +...+\frac{f^{\left( n\right) }\left( x\right) }{\left( n-1\right) !}\left( b-x\right) ^{n-1}+\lambda \frac{\left( b-x\right) ^{n}}{% n!}=\lambda \frac{\left( b-x\right) ^{n}}{n!}-\frac{f^{\left( n+1\right) }\left( x\right) }{n!}\left( b-x\right) ^{n}. \]% Preto\v{z}e $g^{\prime }(c)=0$ m\'{a}me \[ 0=\lambda \frac{\left( b-c\right) ^{n}}{n!}-\frac{f^{\left( n+1\right) }\left( c\right) }{n!}\left( b-c\right) ^{n}\Longrightarrow \lambda =f^{\left( n+1\right) }\left( c\right) . \]% Z predpokladu $g(a)=0$ vypl\'{y}va, \v{z}e \[ 0=f(b)-f(a)-f\,\/\,^{\prime }(a)(b-a)-\frac{f\,\/\,^{\prime \prime }\left( a\right) }{2!}\left( b-a\right) ^{2}-...-\frac{f^{\left( n\right) }\left( a\right) }{n!}\left( b-a\right) ^{n}-f^{\left( n+1\right) }\left( c\right) \frac{\left( b-a\right) ^{n+1}}{\left( n+1\right) !}.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline {\small \hyperref{Sp\"{a}\v{t}}{}{}{M63.tex#1}} \\ \hline \end{tabular} \end{center} \end{document}